Endomorphisms of simple abelian varieties
Posted by Martin Orr on Thursday, 03 January 2013 at 15:51
Today I will discuss the classification of endomorphism algebras of simple abelian varieties.
The endomorphism algebra of a non-simple abelian variety can easily be computed from the endomorphism algebras of its simple factors.
For a simple abelian variety, its endomorphism algebra is a division algebra of finite dimension over .
(A division algebra is a not-necessarily-commutative algebra in which every non-zero element is invertible.)
As discussed last time, the endomorphism algebra also has a positive involution, the Rosati involution.
There may be many Rosati involutions, coming from different polarisations of the abelian variety, but all we care about today is the existence of a positive involution.
Division algebras with positive involutions were classified by Albert in the 1930s.